bls12-381.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "# BLS12-381 sage implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 59,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p in Primes()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "# G1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787"
      ]
     },
     "execution_count": 61,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E1 = EllipticCurve(GF(p), [0, 4])\n",
    "E1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3 * 11^2 * 10177^2 * 859267^2 * 52437899^2 * 52435875175126190479447740508185965837690552500527637822603658699938581184513"
      ]
     },
     "execution_count": 62,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(E1.order())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "E1cofactor = 0x396c8c005555e1568c00aaab0000aaab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3 * 11^2 * 10177^2 * 859267^2 * 52437899^2"
      ]
     },
     "execution_count": 64,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(E1cofactor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "order = 52435875175126190479447740508185965837690552500527637822603658699938581184513"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3006311963586652835656762133460128766163260646573370711678967253222264983490593288693517955057088185894226975257034 : 190532816206323054320525886194094175795290992649002582138245867770654111424551066147978049852962252452029105685383 : 1)"
      ]
     },
     "execution_count": 77,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pa = E1cofactor * E1.random_point()\n",
    "Pa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "assert Pa.order() == order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "# G2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field in i of size 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787^2"
      ]
     },
     "execution_count": 69,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.<I> = GF(p)[]\n",
    "K.<i> = GF(p^2, modulus=I^2+1)\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 + (4*i+4) over Finite Field in i of size 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787^2"
      ]
     },
     "execution_count": 70,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E2 = EllipticCurve(K, [0, 4*(i+1)])\n",
    "E2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "E2order = E2.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "E2cofactor = 0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13^2 * 23^2 * 2713 * 11953 * 262069 * 402096035359507321594726366720466575392706800671181159425656785868777272553337714697862511267018014931937703598282857976535744623203249"
      ]
     },
     "execution_count": 73,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(E2cofactor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
    "assert order*E2cofactor == E2order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3556974607485379506450942071186834848318733278412474767063158158341978728914539372133478039285472233842208924967491*i + 480020083071892448841766069008778524550836399842721415032934076185502765160008572267669855479828669097337168440585 : 2311949564102525834818129998052757219383414673614672803112797707945036274848515620111320569650610401297048611324209*i + 3321138396901700800616131492234377510325735277629746819727785681965062147862156539714231834693308501169920600662845 : 1)"
      ]
     },
     "execution_count": 78,
     "metadata": {
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pb = E2cofactor * E2.random_point()\n",
    "Pb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
   ],
   "source": [
   ]
  }
 ],
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   "language": "sagemath",
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